To may@math.uchicago.edu, categories Sender: categories@mta.ca Precedence: bulk Reply-To: F William Lawvere <wlawvere@buffalo.edu> Probably it was not coined but borrowed, from general topology. For at least 50 years, the two, words chaotic codiscrete were alternate terminology for a certain kind of space. I prefer "codiscrete" since it clearly indicates something opposite to discrete, the precise sense of oppositeness being that of inclusions adjoint to the same uniting functor, often called the "underlying". (In higher dimensions, coskeletal and skeletal are similar identical opposites, with "truncation" as uniter). In fact every groupoid is a colimit of codiscrete ones, indeed groupoids form a reflective subcategory of the topos that classifies Boolean algebras, and the latter has a site consisting of codiscrete groupoids. (The generic Boolean algebra 2^( ) has as its natural geometric realization the infinite-dimensional sphere, containing the ordinary interval as a generating distributive lattice). More recently, "chaotic" has come to have a different meaning, although one also involving a right adjoint. If f:X->Y is a map from a space equipped with an action of a monoid T to another space, then f is a chaotic observable if the induced equivariant map from X to the cofree action Y^T is epimorphic. A classic "symbolic" example has Y=pi0(X), i.e. the observation recorded by f is merely of which component we are passing through, but almost any T-sequence of such is obtained by a sufficiently clever choice of initial state in X. Bill Lawvere
Date: Wed, 28 Sep 2011 20:35:02 -0500 From: may@math.uchicago.edu CC: categories@mta.ca Subject: categories: Reference requested
I have a reference question. Who first coined the term ``chaotic category'' for a groupoid with a unique morphism between each pair of object, and in what context? It is a ridiculously elementary concept, but one that is extremely useful in work on equivariant bundle theory that is needed for equivariant infinite loop space theory and equivariant algebraic K-theory.
Peter May
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]